Improved effective Łojasiewicz inequality and applications (Q6652253)

Łojasiewitz inequality asserts the following: Let \(R\) be a real closed field and \(A\) be a closed and bounded semialgebraic subset of \(R^n\). Let \(f,g: A \to R\) be a semialgebraic continuous functions with \(f^{-1}(0) \subseteq g^{-1}(0)\). Then there exist \(c \in R\) and \(\rho \in \mathbb N\) with \(|g(x)|^\rho \leq c|f(x)|\).\N\NThis paper studies the infimum of \(\rho\) denoted by \(\mathcal L(f,g|A)\) and gives a sharper bound \((8d)^{2(n+7)}\) of \(\mathcal L(f,g|A)\) than the previous studies, where \(d\) is the maximum degree of polynomials defining \(A\), \(f\) and \(g\). A notable feature of this bound is independence of the number of polynomials defining \(A\), \(f\) and \(g\).\N\NApplications of this bound to polynomial optimization are also given. In addition, this paper studies \(\mathcal L(f,g|A)\) for polynomially bounded o-minimal structures and obtains a similar result.